Download Gradient and scattering forces in photoinduced

PHYSICAL REVIEW B 90, 155417 (2014)
Gradient and scattering forces in photoinduced force microscopy
Junghoon Jahng,1 Jordan Brocious,1 Dmitry A. Fishman,2 Fei Huang,3 Xiaowei Li,2 Venkata Ananth Tamma,2
H. Kumar Wickramasinghe,3 and Eric Olaf Potma2,*
1
Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
2
Department of Chemistry, University of California, Irvine, California 92697, USA
3
Department of Electrical Engineering and Computer Science, University of California, Irvine, California 92697, USA
(Received 28 July 2014; revised manuscript received 22 September 2014; published 10 October 2014)
A theoretical and experimental analysis of the dominant forces measured in photoinduced force microscopy is
presented. It is shown that when operated in the noncontact and soft-contact modes, the microscope is sensitive
to the optically induced gradient force (Fg ) and the scattering force (Fsc ). The reconstructed force-distance
curve reveals a tip-dependent scattering force in the 30–60 pN range. Whereas the scattering force is virtually
insensitive to the nanoscopic tip-sample distance, the gradient force shows a z−4 dependence and is manifest
only for tip-sample distances of a few nm. Measurements on glass, gold nanowires, and molecular clusters of
silicon naphtalocyanine confirm that the gradient force is strongly dependent on the polarizability of the sample,
enabling spectroscopic imaging through force detection. The nearly constant Fsc and the spatially dependent Fg
give rise to a complex force-distance curve, which varies from point to point in the specimen and dictates the
image contrast observed for a given set point of the cantilevered tip.
DOI: 10.1103/PhysRevB.90.155417
PACS number(s): 68.37.Ps, 07.79.Lh
I. INTRODUCTION
Atomic force microscopy (AFM) generates images based
on variations in the local forces between an atomic tip and
the surface specimen. The nanoscopic resolution provided by
AFM yields topographic images with exquisite detail, enabling
visualization of individual atoms and intermolecular bonds under optimized conditions [1,2]. Although the contrast in AFM
is primarily topographic in nature, chemical information about
the sample can be obtained by exploiting specific chemical
interactions that may exist in the tip-sample junction [3,4].
Another form of chemical selectively can be obtained by
combining force microscopy with the analytical capabilities
offered through optical spectroscopy.
Dressing the atomic force microscope with spectroscopic
sensitivity is attractive, as it would allow various forms of
molecular spectroscopy to be carried out at high spatial
resolution. A successful example in this category of techniques is infrared (IR) absorption AFM, also referred to as
the photothermal-induced resonance technique [5–8]. In this
technique, the sample is illuminated with infrared light, giving
rise to heating of the material through direct excitation of
dipole allowed vibrational transitions. The thermal expansion
that follows is selective to spectroscopic transitions and can
be probed with high spatial resolution through contact-mode
AFM [9]. IR-based AFM has been used successfully to visualize molecular layers on surfaces with chemical selectivity and
with a spatial resolution down to 25 nm [8].
An alternative method is photoinduced force microscopy
(PIFM), which probes the optically induced changes in the
dipolar interactions between a sharp polarizable tip and the
sample [10,11]. Unlike IR-AFM, the contrast in PIFM does not
rely on the thermal expansion of the material. Rather, the technique is directly sensitive to the electromagnetic forces at play
in the tip-sample junction, and can, in principle, be conducted
in noncontact-mode AFM. The advantage of this approach is
that optical transitions can be probed directly, as dissipation of
excitation energy in the material is not required for generating
contrast. In addition, the photoinduced forces are spatially
confined and fundamentally decoupled from heat diffusion,
giving rise to a high spatial resolution of 10 nm or better [11].
Theoretically, the spectroscopic information probed in this
technique is akin to information accessible through coherent
optical spectroscopy with heterodyne detection [12]. PIFM has
been successfully demonstrated through linear excitation of
dipole-allowed electronic transitions [10] as well as nonlinear
excitation of Raman-active transitions [11].
Although PIFM is a promising technique, the current
theoretical understanding of the relevant forces in the tipsample junction is rudimentary at best. In this contribution,
we provide a simple description of the operating principle
of the photoinduced force microscope in terms of classical
fields and forces. By including both the attractive gradient
interactions and the repulsive scattering force, we simulate the
characteristics of the force-distance curve and compare the
predicted profile with experimental results obtained from gold
nanowires and molecular nanoclusters. Our study identifies
the distance regimes in which attractive gradient forces are
dominant and points out under which conditions molecular
selective signals are optimized. Finally, our experiments show
that the technique can be carried out in the femtosecond illumination mode, paving the way for more advanced nonlinear
optical investigations at the nanoscale level.
II. THEORETICAL DESCRIPTION OF PHOTOINDUCED
FORCE MICROSCOPY
A. Time-averaged Lorentz force
*
Corresponding author: [email protected]
1098-0121/2014/90(15)/155417(10)
We consider a monochromatic electromagnetic wave with
angular frequency ω, which is incident on a polarizable
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©2014 American Physical Society
JUNGHOON JAHNG et al.
PHYSICAL REVIEW B 90, 155417 (2014)
particle. The time harmonic electric and magnetic field
components at location r can be written as
E(r,t) = Re{E(r)e−iωt },
(1)
B(r,t) = Re{B(r)e−iωt }.
(2)
To the first order in the fields, the induced dipole moment of
the particle assumes the same time dependence and is given as
E0
−iωt
μ(r,t)
= Re{μ(r)e
}.
We assume that the particle has no static dipole moment. In
this case, to the first order, the induced dipole moment is
proportional to the electric field at the particle’s position r:
μ(r)
= α(ω)E(r),
t
r
(3)
µp
where α denotes the polarizability of a particle that satisfies
the Clausius-Mossotti equation
−1 3
α = 4π 0
a ,
(4)
+2
FIG. 1. Graphical representation of the symbols used to derive the
mechanical force in the dipolar limit. r denotes the center-of-mass
coordinate.
where a is the particle radius and denotes the complex
dielectric permittivity. Generally, α is a tensor of rank two,
but for atoms and molecules it is legitimate to use a scalar
representation since only the projection of μ
along the
direction of the electric field is of direct relevance.
Using the dipole approximation for calculating the force
between tip and molecule, the cycle-averaged Lorentz force
is [13]
incident electric field and the evanescent field of the induced
dipole in the tip, and vice versa. The mutual interaction can be
modeled by considering both the tip and molecule as spherical
particles of volume 4π
a 3 and an effective polarizability α. In
3
this description, the induced dipole moments in each of the
particles are
F =
α
∇|E|2 + ωα E × B,
2
(5)
where α is a real part of a particle polarizability and α is
an imaginary part of a particle polarizability that satisfies
α(ω) = α (ω) + iα (ω). The first term of Eq. (5) is recognized
as the gradient force (or dipole force) and the second term is
denoted the scattering or absorption force. The gradient force
originates from field inhomogeneities, and is proportional to
the dispersive part (real part) of the complex polarizability.
On the other hand, the scattering or absorption force is
proportional to the dissipative part (imaginary part) of the
complex polarizability.
Equation (5) is a good description of the force if it
can be assumed that the field between the particle and the
tip dipoles varies slowly [13]. This assumption is generally
valid if the particle does not substantially change the spatial
properties of the incident field, as is the case in the point-dipole
approximation. In the case of the gradient force, it has been
shown that the assumption of a slowly varying field produces
acceptable results even when particle and tip dimensions
approach the 10 nm range [13]. Such a spatial scale pertains
to our experiments as well. Therefore, we assume that Eq. (5)
is a valid expression for predicting the general trends of the
photoinduced force as relevant to typical PIFM measurements.
B. Gradient force
The photoinduced gradient force between tip and sample
is a mutual interaction force. The electric field experienced
by the sample particle (molecule) is a combination of the
μ
t = αt (E0 + Ep ),
(6)
μ
p = αp (E0 + Et ),
(7)
where E0 is the electric field of the incident laser, and μ
t and
μ
p are the effective dipole moments of the tip and molecule,
respectively. Et and Ep are the evanescent electric fields of
the tip and molecule, respectively. For our discussion here, the
forces in the z dimension dominate the cantilever response (see
p = μp ẑ,
Figs. 1 and 2). We therefore consider μ
t = μt ẑ and μ
the z components of the dipole moment of the tip and molecule.
Then the static electric fields of the induced dipoles are
Ei =
1 (3μi · r)r − μi
,
4π ε0
r3
Z1+ Z2
Z
(8)
Zc
Sample
FIG. 2. (Color online) Graphical representation of the cantilever
motion. z is the instantaneous position of cantilever. zc is the average
position, z1 is the oscillation of the fundamental resonance, and z2 is
the oscillation of the higher-order resonance.
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GRADIENT AND SCATTERING FORCES IN . . .
PHYSICAL REVIEW B 90, 155417 (2014)
where Ei = Ei x x̂ + Ei y ŷ + Ei z ẑ, r = x 2 + y 2 + z2 , and
i = t,p respectively. Because of the azimuthal symmetry,
Ei x is similar to Ei y . Using (6)–(8), the z component of the
molecule’s dipole moment and its corresponding electric field
Ep can be found as
μp =
2z3 π αp ε0 (αt + 2z3 π ε0 )E0z
,
4z6 π 2 ε02 − αp αt
(9)
αp (αt + 2z3 π ε0 )E0z
,
4z6 π 2 ε02 − αp αt
(10)
Ep z =
with E0 = E0x x̂ + E0y ŷ + E0z ẑ. The expressions (9) and (10)
can be simplified when it is assumed that the distance between
the tip and molecule is larger than the physical size of the
particles, i.e., z > at and z > ap . Because the polarizabilities
of the tip and molecule scale with the cube of the particle
radius, as in (4), we may also write z6 ap3 at3 . Under these
reasonable assumptions, the z component of the gradient force
is reduced to
3αp αt 2
Fg ≡ Fg z −
E .
(11)
2π z4 0z
We see from (11) that the optically induced gradient force
exhibits a z−4 dependence. It is thus a force that is expected
to be prominent in only a narrow nanoscopic range of the
tip-molecule distance, and less relevant for larger separations
of molecule and tip. The negative sign in this interaction
indicates that the force is attractive. Note that the strength of the
force depends on the polarizability αp of the sample particle.
The expression in (11) is similar to the formula previously
2
used in [10]. We will use the parameter β = 3αp αt E0z
/2π
to characterize the magnitude of the gradient force in a given
measurement.
C. Scattering force
Besides the attractive gradient force, Eq. (5) predicts
another optically induced force. This second contribution is
recognized as the scattering force, which is generally repulsive.
The magnitude of the scattering force is proportional to the
dissipative part of the interaction of light with the polarizable
particles, as described by the imaginary part of the complex
polarizability. From Eq. (5), the scattering force can be
described as
Fsc ≡ Fsc z = ωαt E0 × B0 z ,
(12)
where B0 = B0x x̂ + B0y ŷ + B0z ẑ. Only the transverse electric
and magnetic field components contribute to the the force
along the z direction. Given that the dominant component of
the incident field is polarized along x, i.e., E0 ≈ E0x x̂, the
absorption force can be written as
2π αt 2
E0x ,
(13)
λ
where λ is the wavelength of the incident beam. Unlike the
gradient force, the scattering force is repulsive. In addition,
whereas the gradient force is expected to scale as z−4 with
tip-sample distance, the distance dependence of the scattering
force is implicitly contained in the spatial extent of the
excitation field E0x . Because the axial dimension of the
Fsc z =
excitation field is in the (sub-)μm range, the scattering force
manifests itself over a different spatial scale than the local
gradient force. We expect, therefore, that the short-range
nanoscopic interactions are governed by the gradient force,
while at longer distances the scattering force is dominant.
Note that the magnitude of the total photoinduced force
may exhibit a minimum at the tip-sample distance where
the attractive gradient force and the repulsive scattering force
cancel.
D. Cantilever dynamics
We next consider the dynamics of the cantilever-tip system
in the presence of the optically induced force Fopt = Fg + Fsc .
The cantilever motion is modeled as a simple collection of
independent point-mass harmonic oscillators, with motion
confined to the z coordinate only. In practice, two mechanical resonances are of relevance to the photoinduced force
microscopy experiment, namely the first- and a higher-order
mechanical eigenmode of the cantilever-tip system. The first
mechanical resonance is excited by the external driving force
F1 with frequency ω1 , whereas the higher-order resonance at
ωn is actively driven by Fopt . In addition, the cantilever tip is
subject to the interaction force Fint , which includes all relevant
van der Waals, electrostatic, magnetic, chemical, and Casimir
forces, as well as average (unmodulated) optical forces. In this
model, the equations of motion for the displacement of the
first- (z1 ) and higher-order (zn ) mode are found as
mz̈1 + b1 ż1 + k1 z1 = F1 cos(ω1 t) + Fint (z(t)),
(14)
mz̈n + bn żn + kn zn = Fopt (z(t)) cos(ωn t) + Fint (z(t)), (15)
where m is the effective mass of the cantilever; ki and bi are,
respectively, the force constant and the damping coefficient
of the ith eigenmode [14]. In the following, we will set
n = 2, corresponding to the tip displacement of the second
mechanical resonance, but the analysis applies equally to
higher-order modes. Note that the resonance frequency ωi
and
√ the external driving force F1 can be expressed as ωi =
ki /m and F1 = k1 A01 /Q1 , respectively, where A01 is the
free oscillation amplitude of the driven first eigenmode of the
cantilever, and Qi is the quality factor of the ith eigenmode.
In addition, the damping coefficient can be expressed as bi =
mωi /Qi . The instantaneous tip-surface distance is represented
by
z(t) = zc + z1 (t) + z2 (t) + O(ε)
≈ zc + A1 sin(ω1 t + θ1 ) + A2 sin(ω2 t + θ2 ), (16)
where zc is the equilibrium position of the cantilever, Ai is
the amplitude and θi is the phase shift of the ith eigenmode,
and O(ε) is a term that carries the contribution of the other
modes and harmonics. Substituting Eq. (16) into Eqs. (14)
and (15) by multiplying both sides of the resulting equation
by sin(ωi t + θi ) and cos(ωi t + θi ), followed by an integration
over the oscillation period, the following general relations
for amplitude, phase, mechanical interaction force, and
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JUNGHOON JAHNG et al.
PHYSICAL REVIEW B 90, 155417 (2014)
photoinduced force are obtained:
A1
F1
k1 − mω1 2
=
sin θ1 +
2
2
T
Fint sin(ω1 t + θ1 )dt,
0
(17)
T
b1 ω1 A1
F1
Fint cos(ω1 t + θ1 )dt, (18)
=
cos θ1 +
2
2
0
T
2 A2
k2 − mω2
=
[Fint +Fopt cos(ω2 t)] sin(ω2 t + θ2 )dt,
2
0
(19)
T
b2 ω2 A2
[Fint + Fopt cos(ω2 t)] cos(ω2 t + θ2 )dt. (20)
=
2
0
If the form of Fint and Fopt is known, the amplitudes A1 ,A2
and the phase shifts θ1 ,θ2 can be calculated through numerical
integration of Eqs. (17)–(20).
the general physics of the problem remains preserved. Using
the approximation ω2 ≈ 6ω1 , the forces are simplified to
∞
F1
sin θ1 (z) − k1 − mω12 dz, (24)
Fc (z) =
A1 (z)
z
F1
cos θ1 (z) − b1 ,
A1 (z)ω1
2
|Fopt (z)| = A2 (z) m2 ω 22 − ω22 + b 22 ω22 ,
(z) =
with
ω2 =
k2 −
∂Fc m,
∂z z
b 2 = b2 − (z).
(25)
(26)
(27)
The formalism outlined by Eqs. (24)–(26) makes it possible
to reconstruct the distance-dependent mechanical and photoinduced force from experimentally accessible parameters.
E. Reconstruction of distance-dependent force
The amplitude and phase of the first and second resonances
are experimentally accessible quantities. We next describe an
approximate method to relate the forces active at the tip-sample
junction to these experimental quantities. In general, Fint is
a nonlinear function of the tip-sample distance z and contains both conservative and dissipative (i.e., nonconservative)
forces,
Fint = Fc (z) + Fnc ,
(21)
where the conservative term Fc , by definition, depends only
on the distance z. For our description here, we assume that
the nonconservative force can be written as Fnc = −
(z)ż,
where represents the effective damping coefficient of a
given dissipative interaction. Such a form of force describes
many physical interactions that give rise to an energy loss
of the cantilever movement [15–17]. Note that although
the term −
(z)ż may be replaced by some other specific
forms of dissipation, one can still employ the effective and
intuitive coefficient as a parameter describing a given
dissipation [18,19].
We use two approximations to simplify the description.
First, we assume that the tip displacement is small such that
the force at z can be obtained through a Taylor expansion
of the force at the equilibrium position zc [20]. Under
these conditions, the mechanical interaction force and the
photoinduced force can be expressed by
Fint (z(t)) = Fc (z) − (z)ż
∂Fc (z) (z − zc ) − (zc )ż, (22)
∂z zc
∂Fopt Fopt (z(t)) ≈ Fopt (zc ) +
(z − zc ).
(23)
∂z zc
≈ Fc (zc ) +
Second, we assume that the frequency of the first and second
mechanical resonance are related as ω2 ≈ 6ω1 . In practice, the
frequency of the second resonance is 6.27ω1 [14]. Although the
approximation ω2 ≈ 6ω1 gives rise to numerical differences in
the integration of Eqs. (17)–(20), this difference is small while
III. EXPERIMENTAL METHODS
A. Light source
The reported experiments are carried out with a femtosecond light source. The pulsed light is derived from a ti:sapphire
laser (MaiTai, Spectra-Physics), which delivers 200 fs pulses
with a center wavelength of 809 nm. The pulse repetition
rate is 80 MHz and the average power of the laser beam at
the sample plane is between 60 μW and 92 μW, depending
on the experiment. The laser light is precompressed with a
prism compressor to account for dispersion in the microscope
optics. In addition, the laser light is amplitude modulated at
a frequency that coincides with a higher-order mechanical
resonance of the cantilever used in the experiment, as indicated
below.
B. Atomic force microscope
A custom-modified atomic force microscope (Molecular
Vista) is used for the photoinduced force experiments. A
simplified scheme is given in Fig. 3. The system consists of
an inverted optical microscope outfitted with an NA = 1.40
oil immersion objective, a sample stage scanner, an AFM
scan head, and transmission and reflection optics for sample
inspection. The microscope objective generates a diffractionlimited focal spot at the sample surface using x-polarized laser
light. In the focal plane, significant portions of y-polarized
and z-polarized light are present, which have distinct focal
field distributions. The tip is positioned at the location of
maximum z-polarized light [21] to enhance the sensitivity of
the measurement to the z-directed gradient forces.
A 30 nm radius gold coated silicon tip is used in
the experiments (FORTGG, AppNano). Two cantilever systems are employed. The first cantilever system has k1 =
1.61 N/m, Q1 = 181.93, and a first mechanical resonance
f01 = 58.73 kHz. The second mechanical resonance of the
cantilever is at f02 = 372.41 kHz, with k2 = 52.24 N/m
and Q2 = 561.36. The free oscillation amplitude of the
fundamental resonance A01 is set to 38 nm in the experiments
below. The laser modulation frequency was set to 372.41 kHz
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PHYSICAL REVIEW B 90, 155417 (2014)
by spin-casting a concentrated solution of SiNc in toluene onto
the plasma-cleaned coverslip. After evaporation of the solvent,
nanoscale aggregates of SiNc are formed, varying in size from
several micrometers to less than 10 nm in diameter.
IV. RESULTS AND DISCUSSION
A. Force-distance simulations
We first study the general trends of the cantilever dynamics
in the presence of both mechanical and photoinduced forces
using Eqs. (17)–(20). For this purpose, it is required to choose
a functional form of the forces Fint and Fopt that are active at
the tip-sample junction. We assume that Fint is described by
a conservative Lennard-Jones-type force that contains a 1/z6
(repulsive) and a 1/z2 (attractive) force term,
4
l
1
(28)
Fint = f0
− 2 ,
3z6
z
where f0 is a constant and l is the characteristic distance at
which the attractive force is minimized. For Fopt we obtain
from Eqs. (11) and (13)
Fopt = −
FIG. 3. (Color online) Sketch
microscope.
of
the
photoinduced
force
for photoinduced force measurements, coinciding with the
second mechanical resonance of the cantilever. The second
cantilever system exhibits a mechanical resonance at f01 =
58.58 kHz with k1 = 1.6 N/m and Q1 = 178, operated with
a free oscillation amplitude A01 = 41 nm. The photoinduced
force measurements with this system are carried out at the
third mechanical resonance of the cantilever, which is found
at f03 = 1033.190 kHz. The corresponding spring constant
and quality factor of this resonance are k3 = 492.51 N/m and
Q3 = 539, respectively.
The topographic and PIFM images are acquired simultaneously. This is achieved by demodulating the tip response at f01
to retrieve the topography image and at f02 (or f03 ) to retrieve
the photoinduced force image. The AFM controller of the
Molecular Vista system includes a digital lock-in amplifier,
used for demodulation of the signals, and a digital function
generation with a field programmable gate array (FPGA), used
for generating synchronized clock signals.
C. Sample materials
Two test samples are used in this study. The first sample
is a composed of a 0.17 mm thick borosilicate coverslip
with deposited gold nanowires. The nanowires are fabricated
by the lithographically patterned nanowire electrodeposition
technique [22]. The resulting nanowires feature an average
width of 120 nm and an average height of 25 nm, whereas the
length of the nanowires extends over millimeters. The second
sample is a borosilicate coverslip with deposited nanoclusters
of silicon naphtalocyanine (SiNc). The samples are prepared
3αt αp
2π z4
2
E0z
+
2π αt 2
E0x .
λ
(29)
Figure 4 shows the distance-dependent amplitude (a) and
the phase (b) of the first resonance, as well as the amplitude
of second resonance (c) as simulated with Eqs. (17)–(20)
and the chosen forces above. The z-dependent amplitude and
phase of the first resonance show the expected profile for the
chosen form of Fint . The amplitude of the second resonance,
on the other hand, displays a very different profile, depicted
by the black curve in Fig. 4(c). A peak near z ≈ 11 nm is
seen, the amplitude of which is dependent on the magnitude
of the effective polarizability. Around z ≈ 15 nm a sharp
dip is observed, followed by a plateau at longer distances.
Unlike the first resonance, the second resonance is sensitive
to the presence of Fopt . The characteristic shape of the
amplitude-distance curve reflects the presence of both the
attractive gradient force and the repulsive scattering force. At
shorter distances, the gradient force dominates, whereas for
longer tip-sample separations the response is governed by the
scattering force. The dip appears when the two optical forces
cancel, i.e., Fg + Fsc = 0.
The red curve in Fig. 4(c) is obtained by increasing the
magnitude of Fg by a factor of 2; i.e., β is twice as large as for
the black curve, whereas Fsc remains the same. Two immediate
changes are observed. First, the maximum amplitude A2 is
significantly larger for the 2β case, as can be expected for
an increase in the magnitude of the gradient force. Second,
the cancellation of the two optical forces now occurs at a
longer distance, near ∼18 nm rather than near ∼15 nm in this
simulation. An increase in the gradient force thus gives rise
to an amplitude-distance curve that appears shifted to longer
distances.
The formalism outlined in Eqs. (24)–(26) can be used to
reconstruct the mechanical and photoinduced forces between
the tip and sample from the simulated amplitude-distance
curves. The black solid line in Fig. 4(d) represents the
functional form of the assumed mechanical force, as given
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JUNGHOON JAHNG et al.
A1 (nm)
(a)
PHYSICAL REVIEW B 90, 155417 (2014)
by Eq. (28). The purple dots indicate the reconstructed force
from the simulated measurement. The close correspondence
between the assumed and reconstructed data indicates that
the assumptions made in (24)–(26) are justified. Similarly, the
solid black line in Fig. 4(e) shows the chosen form of the
photoinduced force, described by Eq. (29), whereas the purple
dots correspond to the reconstructed data. The simulated experiments presented in Fig. 4 underline that the photoinduced
force can be extracted from measured amplitude-distances
curves.
The red line in Fig. 4(e) corresponds to the situation where
β is twice as large as for the black solid line. Similarly
to Fig. 4(c), the point where Fg and Fsc cancel lies at
a longer distance. The shift of the cancelation point is a
direct manifestation of the situation where Fg increases while
Fsc remains constant. The set point of the cantilever thus
has a profound influence on the relative magnitude of the
measured optical forces. Consider the two curves in Fig. 4(e),
representing an object in the sample with β (black) and a
second object with 2β (red). Depending on the set point of the
cantilever, the contrast between these two objects may vary
drastically. For the values used in this simulation, if the set
point is at ∼15 nm the force measured at the object represented
by the black curve is minimum while the net photoinduced
force measured at the object with the red curve is substantial.
However, if the cantilever set point is moved to 20 nm, the
situation is reversed. The strongest force is now measured at
the object with the weaker gradient force, as the net force at
this distance is governed by Fsc . Hence, objects which give rise
to a stronger Fg do not always produce the strongest signals in
PIFM. The contrast observed is dependent on the cantilever set
point. Consequently, knowledge of the force-distance curve is
essential for interpreting images generated with photoinduced
force microscopy.
20
10
0
(b)
40
1
(degree)
80
0
(c)
600
2
A2 (pm)
400
200
0
(d)
Mechanical
force (pN)
80
40
B. Photoinduced forces near gold nanostructures
0
Photo-induced
force (pN)
(e)
40
2
20
0
0
10
20
30
Tip-sample distance (nm)
FIG. 4. (Color online) Simulation of the amplitude-distance
curve (a) and the phase-distance curve (b) of the fundamental
resonance of a cantilever. The free oscillation amplitude of A01 is
20 nm. (c) Amplitude-distance curve of the second resonance of
the cantilever for Fg with β (black) and 2β (red). (d) Pre-assumed
Lennard-Jones-type force (solid black) and the reconstructed force
(purple dots). (e) Pre-assumed total magnitude of the optical force
for β (solid black) and 2β (solid red), along with the corresponding
reconstructed forces for β (purple dots) and 2β (green triangle). The
experimental parameters of the simulation are given in [23].
We next apply the developed formalism to experimentally
obtained amplitude-distance measurements. Gold nanowires
deposited on a glass surface are chosen as the test object.
Figure 5 shows the amplitude (a) and phase (b) of first
resonance and the amplitude of second resonance (c), which
are recorded simultaneously while the tip is parked over
the gold nanowire. The amplitude of the second resonance
(A2 ) shows the characteristic peak, followed by a shallow
dip and a plateau at longer distances. At shorter tip-sample
distances, A2 settles at ∼195 pm, as the tip enters hard-contact
mode, indicated by the blue data points. As expected, in this
regime A1 and θ1 remain invariant. This region is relevant to
photothermal force measurements [5–8].
Figure 5(d) depicts the reconstructed force from the
measured amplitude and phase curves (black squares). The
extracted curve shows three clear regimes. At short distances,
in the hard-contact region, the force is invariant. As the set
point of the cantilever is decreased, the gradient force becomes
important. The observed dip corresponds to the distance
where Fg + Fsc = 0. At longer distances, the scattering force
dominates. Note that the force does not approach exactly zero
at the dip. The observed offset is attributed to the presence
of thermal noise at the mechanical frequency ω2 , resulting
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GRADIENT AND SCATTERING FORCES IN . . .
PHYSICAL REVIEW B 90, 155417 (2014)
0.6
(a)
(a)
ext=
809 nm
Absorbance
A1 (nm)
40
20
600
700
Wavelength (nm)
13 nm
(b)
0
0.0
900
580 µV
(c)
(b)
0
40 nm
1
(degree)
70
40 nm
0 nm
290 µV
-70
(c)
0
Line cut (nm)
45
0.5
450
(e)
A2 (µV)
Height (nm)
3
(d)
4nm
0
Line cut (nm)
45
310
250
A2 (pm)
FIG. 6. (Color online) (a) Linear absorption spectrum of siliconnaphtalocyanine (SiNc). The laser spectrum is indicated by the
shaded area. (b) Topography image of clusters of SiNc and
(c) the corresponding photoinduced force image of SiNc. (d) Onedimensional cut at the line shown in the topography image (b).
(e) One-dimensional cut at the line shown in the photoinduced force
image (c).
200
Photo-induced
force (pN)
(d)
C. Photoinduced forces and molecular resonances
120
60
0
-40
-20
0
Tip-sample distance (nm)
FIG. 5. (Color online) The amplitude (a) and the phase (b) of the
fundamental resonance with respect to tip-sample distance on gold
nanowire. The zero is the set point at which the tip is engaged.
The tip is retracted from the sample. (c) The amplitude of the
second resonance of the cantilever on the gold nanowire. (d) Squares
represent the reconstructed experimental data from the nanowire and
the red line is the model fit. Blue data points indicate the hard-contact
region.
in a residual fluctuation force [24,25]. The reconstructed
force curve can be fitted by Eq. (29), which is shown by
the red line in Fig. 5(d). Based on the fit, an estimated
value for the scattering force can be extracted as Fsc =
32.0 pN, and gradient force prefactor β is found as 2.06 ×
10−42 N m4 .
We next apply PIFM with fs pulses to visualize nanoscopic
clusters of molecules. The sample used in this study is
silicon naphtalocyanine (SiNc) molecules deposited on a
glass coverslip. SiNc displays a strong absorption at 800 nm,
which significantly overlaps with the spectrum of the laser
pulses used. Figure 6(b) shows the topography image of a
nanoscopic cluster of SiNc molecules on the glass surface. The
corresponding photoinduced force image is shown in Fig. 6(c).
The PIFM image shows the same cluster as identified in the
topography image. Whereas the topography image suggests
height variations in the cluster, the PIFM image shows a nearly
uniform signal in the cluster. This response is expected from
the gradient force, which is a local force that is manifest close
to the surface and relatively independent of volume effects. In
addition to the larger cluster, smaller structures can be seen
in Fig. 6(c) which appear absent in the topography image.
The one-dimensional line cuts shown in Figs. 6(d) and 6(e)
confirm that some smaller features are uniquely resolved in
the photoinduced force image. The width of these structures
is <10 nm, corresponding to the spatial resolution of the
photoinduced force microscope. Similar structures have been
seen before in photoinduced force microscopy, and have been
suggested to originate from a single or a few molecules [10].
Figures 7(a) and 7(b) show the amplitude- and phasedistance curves, respectively, when the illuminated tip is positioned on the large cluster of molecules. These measurements
155417-7
JUNGHOON JAHNG et al.
PHYSICAL REVIEW B 90, 155417 (2014)
Photo-induced force (pN)
(a)
A1 (nm)
30
20
10
0
(b)
200
SiNc
SiNc fit
glass
glass fit
100
0
0
120
A2 (pm)
Photo-induced
force (pN)
30
40
40
80
(d)
20
FIG. 8. (Color online) Comparison of the retrieved forcedistance curves measured on SiNc and on glass.
0
(c)
10
Tip-sample distance (nm)
1
(degree)
80
200
100
0
-20
0
20
Tip-sample distance (nm)
FIG. 7. (Color online) The amplitude (a) and the phase (b) of
the fundamental resonance with respect to tip-sample distance on
Si-naphthalocyanine (SiNc). The zero is the set point at which the tip
is engaged. The sample is on the negative side. The tip is retracted
from the sample. (c) The amplitude of the second resonance of the
cantilever on SiNc. (d) The reconstructed optical forces on SiNc
(squares) and the model fit (solid red line).
help identify the onset of tip engagement and the regime in
which hard contact (blue data points) is reached. Photoinduced
force measurements are performed by overlapping the laser
modulation frequency with the third mechanical resonance
of the cantilever at f03 = 1033.190 kHz. At this higher frequency, the effects of fluctuating thermal force contributions is
expected to be reduced. The amplitude of the third mechanical
resonance is shown in Fig. 7(c) and the extracted force is
plotted in Fig. 7(d). Similar to the measurements on the gold
nanowire, a clear dip can be seen in the force-distance curve,
which is attributed to the cancellation of Fg and Fsc . Using
Eq. (29), the force-distance curve can be fitted. From the fit,
we find that Fsc = 59.0 pN and β = 6.10 × 10−43 N m4 . Note
that the β prefactor in the SiNc measurements is smaller than
in the measurements on gold, indicating that the gradient force
between the tip and the SiNc is weaker than for the case of the
gold nanowire.
In Fig. 8 we compare the force-distance curve measured on
another SiNc cluster with the force-distance curve recorded on
the glass surface adjacent to the cluster. We observe that the
cancellation point between Fg and Fsc appears for shorter tipsample distances on the glass surface compared to when the tip
is placed over the SiNc cluster. This is a consequence of the
stronger Fg measured over the molecular cluster. To cancel out
the stronger attractive force, the tip has to be moved farther
from the sample, yielding an apparent shift of the dip in the
force-distance curve.
The hard-contact region, where thermal expansion effects
are most prominently measured, is clearly identified in
the measurement. Fitting of the curves yields β = 1.64 ×
10−42 N m4 for the SiNc cluster and β = 1.75 × 10−43 N m4
for the glass surface. The stronger gradient force for the
molecular cluster is expected, as the molecular resonance
dresses the material with a higher polarizability than the
transparent glass material. The gradient force, therefore, is
a sensitive probe of the optical properties of the sample under
the tip. The scattering force in both measurements, however,
is comparatively similar: Fsc = 52.1 pN for the SiNc cluster
and Fsc = 45.3 pN for the glass surface. Unlike the gradient
force, the scattering force extends over a larger scale and is
predominantly dictated by the optical response of the tip. The
scattering force is relatively insensitive to the polarizability of
the (molecular) sample, which is also evidenced by Eq. (13).
D. Feasibility of photoinduced force microscopy with fs pulses
Unlike previous photoinduced force experiments, which
were carried out with cw illumination [10,11], the force
measurements reported here are conducted with 200 fs optical
pulses derived from a ti:sapphire laser. Because of the much
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GRADIENT AND SCATTERING FORCES IN . . .
PHYSICAL REVIEW B 90, 155417 (2014)
higher peak powers compared to cw irradiation, the use of fs
pulses may raise concerns regarding optical nonlinearities at
the tip that may negatively affect the stability and duration
of the experiment. It is known that gold-coated tips exhibit
plasmon resonances that can be driven at the excitation
wavelength of 809 nm, which give rise to strong fields at the
apex [26,27]. Such high fields may induce nonlinear optical
photodamage of the tip’s nanomorphology, which in turn
affects the effective polarizability of the tip and the optical
force that results from it. In addition, unfavorable heat kinetics
can give rise to time-varying thermal fluctuations on the time
scale of the experiment, which may overwhelm the cantilever
dynamics driven by the pN photoinduced forces.
However, our experiments suggest that photoinduced force
measurements can be confidently conducted with fs illumination. We find that keeping the illumination power well
below 100 μW sustains stable conditions that enable force
measurements over several hours. The use of a high-repetition
rate laser is favorable in this regard, as a steady state in the
tip-heating dynamics is achieved on the relevant time scale of
the experiment. The ability to use fs pulsed excitation in optical
force measurements opens exciting prospects for developing
experiments where the measured force is induced by selected
nonlinear optical manipulations of the sample.
V. DISCUSSION AND CONCLUSION
from point to point. Therefore, the positive (attractive) contrast
in photoinduced force microscopy is dictated by the spatial
variation of Fg relative to Fsc .
Our formulation allows the reconstruction of the photoinduced forces from amplitude-distance measurements. Experiments on bare glass, gold nanowires, and molecular clusters
confirm that Fg and Fsc manifest themselves on different
spatial scales. Fits to the reconstructed force-distance curves
indicate that Fg is local and, because of the z−4 dependence,
significant only for tip-sample distances within a few nm. In
addition, Fg is sensitive to the polarizability of the sample
object. For instance, the polarizability of the glass surface at
the 809 nm excitation wavelength is significantly lower than
the polarizability of the optically resonant SiNc nanocluster,
resulting in a much stronger measured Fg for the latter.
The scattering force, on the other hand, shows no distance
dependence on the nanoscale. This observation complies with
the theoretical description of Fsc in Eq. (13), which states that
it is manifest over a length scale defined by the axial extent of
the optical excitation field (∼μm).
Whereas Fg varies dramatically with the polarizability
of the sample, the measurements, conducted under similar
experimental conditions, reveal that Fsc is in the 30–60 pN
range for all measurements presented here. This is to be
expected from Eq. (13), which indicates that the magnitude
of the scattering force is defined by the interaction of the
excitation light with the tip. Naturally, the magnitude of Fsc
is a direct function of the tip material and morphology. The
different distance dependence and polarizability dependence
of Fg and Fsc gives rise to a force-distance curve that can vary
dramatically from point to point. The contrast in the image
is also dependent on the set point of the cantilever. Changing
the set point may change the contrast of a given object on the
glass surface from positive to negative relative to the signal
from the glass. This notion emphasizes that understanding the
mechanisms at play in PIFM is essential for interpreting the
images produced with this technique.
In short, we have theoretically and experimentally characterized the dominant forces in photoinduced force microscopy. The framework developed here makes it possible
to quantitatively extract Fg and Fsc from photoinduced force
measurements, which is essential for interpreting the image
contrast. Understanding the contrast mechanism in PIFM is
important for relating the photoinduced forces to molecular
spectroscopic information.
In this work, we have studied the mechanisms that
contribute to the signals measured in photoinduced force
microscopy. This new form of microscopy operates in a regime
away from the hard-contact mode, as is required in IR AFM
techniques. Consequently, thermal expansion of the sample
after optical excitation is not the primary contrast mechanism
in PIFM. Instead, the technique is directly sensitive to the
electromagnetically induced forces in the tip-sample junction,
as described by the gradient force Fg and the scattering
force Fsc . PIFM enables the measurement of Fg and Fsc , and
generates images with contrast based on the spatial variation
of these two optical forces.
We have provided a theoretical description of cantilever
dynamics in the presence of Fopt , the sum of Fg and Fsc . The
photoinduced force is conveniently detected at modulation
frequencies that coincide with higher-order resonances of
the cantilever system. The amplitude of the cantilever at the
frequency of the higher-order mechanical resonance reflects
the competition between the attractive gradient force and the
repulsive scattering force. The simulated amplitude-distance
curve reveals a minimum at the tip-sample distance where
Fg and Fsc cancel. The z-distance at which the minimum
occurs may change as a function of lateral position in the
sample, as the relative contributions of Fg and Fsc may vary
We thank Jonathan Burdett for his experimental help during
the early stages of this project. This work was supported by
the National Science Foundation, Grant No. CHE-0802913.
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